3 Answers
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First attested in the 1960s in the context of aviation. Probably coined by pilots whose throttle levers had round, ball-like tops and for whom putting the "balls to the wall" (the firewall of the aircraft) meant making the aircraft fly as quickly as possible.

Actually, the balls are in a fairly neutral position in the main photo, and they are pushed toward the firewall in the control quadrant close-up. But the reason for the photo is to show you that planes can have two throttle levers.
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WinnieNicklausFeb 20 '14 at 18:48

3

While this may be true, I have a hard time believing that the phrase does not refer to testicles. This etymology sounds sanitized.
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Bradd SzonyeFeb 20 '14 at 22:42

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It doesn't have to be one or the other -- while the literal interpretation of the phrase would refer to the throttle levers and firewall, giving a coherent meaning to the phrase, it's entirely possible that it only caught on because of the double entendre.
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WinnieNicklausFeb 20 '14 at 22:47